If you are considering a Comedy structurally, then neither Puck nor Bottom is the protagonist.  Both are characters whose actions (Puck) contribute to the complications of the main plot or whose actions (Bottom) comprise a subplot that comments on the characters/events of the main plot.  Since neither's actions are actually the driving force of the main plot, neither can be considered the protagonist.  So, who then, is the protagonist?

Classically, a comedy must end in at least one marriage, and it is the central character(s), the protagonist(s) who go(es) through the main trials and tribulations of the play, only to have all ironed smoothly out for the "happy" ending in marriage.

One such charcter is Hermia.  She begins the play in a dilemma (similar to a dilemma Juliet faces in Romeo and Juliet).  She wants to marry for love to Lysander, but her father wants her to marry Demetrius.  She is given an order to marry Demetrius or else suffer the consequences -- be killed (!!) or join a nunnery.  She decides to flee to the forest with Lysander.

There is also a case to be made for Helena as a protagonist, since she is in love with Demetrius at the beginning of the play, and creates a plan to alert him to Lysander and Hermia's scheme and  then follow him, hoping to win back his love.

A great deal of the mischief and mix-up in the play revolves around the young lovers' story lines.  As a group, they could be considered the protagonists, or you could make a case, as I have done above, for one of them.  And, as is traditional in Shakespeare's comedies, there is a subplot invoving low-born characters, the clowns, who are in the play to entertain the audience in the very ways you describe.  They are often the ones who "steal the show."  But this does not make them the protagonists of the story.

Bottom is the foil, Puck is the classic trickster. The trickster character is generally a messenger to the gods or for the gods whose amorality and desire to "play" often serve as the catalyst for events that are set into motion. These, in drama, are usually humorous in nature such as the mistaken identities and mixed up outcomes that prevail in "Midsummer Night's Dream".  Puck is the driving force for all of the action as well as the conscience of the characters (both the good side and the bad side). As such, it is definitely possible to argue for is role as the protagonist around whose actions the story revolves. As he closes the play, his final monologue begs the audience's forgiveness and wishes them well for having been participant's in Pucks dream or vision, so again it all springs forth from Puck.

There is a good bit of evidence to support the premise that Puck is the protagonist in Shakespeare's Midsummer Night's Dream, despite the fact that Bottom does steal the show. However, if Puck had not decided to turn Bottom into an ass in the first place, he might never have caught the attention of the Fairy Queen and had the opportunity to be such an important player in the affairs of Oberon and Titania's court.  Both Puck and Bottom are different than the other characters in that they actually interact with characters in both the fairy and mortal realms.  Puck's actions drive the plot as soon as the humans intrude on the fairy gathering in the forest; he is Oberon's henchman, but he acts independently as well, and he is the comic mover of the action of the play.

Moreover, Shakespeare gives Puck the final speech in the epilogue, which is usually reserved for the main protagonist,  for example, Rosalind in As You Like It, another indication of his importance in the play.

Why may Puck be considered the protagonist in A Midsummer Night's Dream by William Shakespeare and not Bottom?

A Midsummer Night's Dream by William Shakespeare does not have a traditional protagonist. Normally, a protagonist is both the most important character in a play and the person whose desires, needs, or situation drives the plot of the play. The main conflict of the play is usually concerned with how the protagonist struggles against obstacles to reach a goal or resolve a problem, sometimes successfully and sometimes not.

The main conflicts in the play are among the two sets of young lovers and Titania and Oberon. In the case of the young lovers, we have a fairly conventional narrative trajectory in which Hermia loves Lysander and Helena loves Demetrius but there are obstacles to their happy unions; after a few plot twists, both young women end up with the men they love. Thus in this sense we have a very traditional comic plot arc in the lovers' stories, and in some ways the young lovers are the closest characters to traditional protagonists.

The Titania-Oberon conflict over the changeling is also resolved, but we do not have a fully fleshed out narrative arc, meaning that they do not function as protagonists.

Bottom, as one of the rustics, functions as comic relief, and although he is part of Titania's story, his role, and that of the play within the play, are comic diversions; he has no major narrative of his own about his own aspirations and goals.

Puck is a more important character in the play than Bottom, but carries out his deeds at the behest of Oberon. We don't really get a sense of Puck as having motives and a story of his own, but merely acting in and recounting the stories of others. In a way, he is like the playwright, a creator and facilitator of other people's stories rather than the protagonist in his own story.  

Though Bottom often steals the show in performance, Puck is usually considered the most important character in A Midsummer Night’s Dream. Comparing Puck to Bottom, why might Puck be considered the protagonist?

Puck has a big supporting role in Shakespeare'sA Midsummer Night's Dream, but saying that he's the most important character in the whole play seems to be stretching it a bit. When considering importance, a character's influence on the plot of the story and upon other characters must certainly be taken into account. Puck acts at the request of Oberon, so one might argue that Puck himself isn't the brains behind the thickening of the plot. However, he does make a mistake when he casts the love spell on Lysander instead of Demetrius and the lovers are twisted up into more confusion. Bottom is the main character for the play within the play, but he's also made fun of and treated without respect by everyone. Bottom is also the victim of Oberon's  manipulation of Titania which does not prove him to be a hero of any type. Hence, if Puck and Bottom were in a contest to be crowned "protagonist," then the award would have to go to Puck because he is not a victim; and at the end of the play he attempts to smooth things over with the audience if they were offended, which shows that he is more pro-active that Bottom.